Question

Check whether $ {(x + 3)^3} = {x^3} - 8 $ is a quadratic equation?

Answer 1

Hint: First of all take the given expression and move all the terms on one side of the equation and compare the resultant equation with the standard equation of the quadratic which is $ a{x^2} + bx + c = 0 $

Complete step-by-step answer:
Take the given expression: $ {(x + 3)^3} = {x^3} - 8 $
Expand the solution left using the identity of the algebraic expression $ {(a + b)^3} = {a^3} + {b^3} + 3ab(a + b) $
 $ {x^3} + {3^3} + 3x(3)(x + 3) = {x^3} - 8 $
Simplify the above expression –
 $ {x^3} + {3^3} + 9x(x + 3) = {x^3} - 8 $
Terms outside the bracket are multiplied with all the terms inside the bracket.
 $ {x^3} + {3^3} + 9{x^2} + 27x = {x^3} - 8 $
Place the value for the cube of the terms, cube is the number multiplied thrice for example $ {3^3} = 3 \times 3 \times 3 = 27 $
 $ {x^3} + 27 + 9{x^2} + 27x = {x^3} - 8 $
Move all the terms on one side of the equation, when you move any term from one side of the equation to the opposite side of the equation then the sign of the terms also changes. Positive terms become negative and the negative term becomes positive.
 $ {x^3} + 27 + 9{x^2} + 27x - {x^3} + 8 = 0 $
Combine like terms together in the above equation –
 $ \underline {{x^3} - {x^3}} + 9{x^2} + 27x + \underline {8 + 27} = 0 $
Simplify the like terms in the above expression. When you combine like terms with the same value and opposite sign then they cancel each other.
 $ 9{x^2} + 27x + 35 = 0 $
Comparing the above resultant equation with the standard quadratic equation $ a{x^2} + bx + c = 0 $ , it implies that the given equation is the quadratic equation.
So, the correct answer is “ $ 9{x^2} + 27x + 35 = 0 $ , It is Quadratic equation”.

Note: Be careful about the sign convention while simplifying the given equation. When you move any term from one side to the opposite side then the sign of the term changes, positive changes to negative and the negative changes to the positive. Remember the different identities of squares and cubes to get the simplified form.